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Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Author

Listed:
  • Dario Bauso

    (The University of Sheffield
    Università di Palermo)

  • Quanyan Zhu

    (Polytechnic School of Engineering New York University)

  • Tamer Başar

    (University of Illinois at Urbana-Champaign)

Abstract

Mixed integer optimal compensation deals with optimization problems with integer- and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type of approximation and extend our study to opinion dynamics in social networks as a special case of interest.

Suggested Citation

  • Dario Bauso & Quanyan Zhu & Tamer Başar, 2016. "Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 606-630, May.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-016-0881-6
    DOI: 10.1007/s10957-016-0881-6
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    References listed on IDEAS

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    1. Hamidou Tembine & Quanyan Zhu & Tamer Basar, 2011. "Risk-sensitive mean field stochastic differential games," Post-Print hal-00643547, HAL.
    2. POCHET, Yves & WOLSEY, Laurence A., 1993. "Lot-sizing with constant batches: formulation and valid inequalities," LIDAM Reprints CORE 1066, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Sebastian Sager & Mathieu Claeys & Frédéric Messine, 2015. "Efficient upper and lower bounds for global mixed-integer optimal control," Journal of Global Optimization, Springer, vol. 61(4), pages 721-743, April.
    4. Andrew J. Clark & Herbert Scarf, 2004. "Optimal Policies for a Multi-Echelon Inventory Problem," Management Science, INFORMS, vol. 50(12_supple), pages 1782-1790, December.
    5. Yves Pochet & Laurence A. Wolsey, 1993. "Lot-Sizing with Constant Batches: Formulation and Valid Inequalities," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 767-785, November.
    6. Daron Acemoğlu & Giacomo Como & Fabio Fagnani & Asuman Ozdaglar, 2013. "Opinion Fluctuations and Disagreement in Social Networks," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 1-27, February.
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    Cited by:

    1. Stefanny Ramirez & Dario Bauso, 2021. "Dynamic Coordination Games with Activation Costs," Dynamic Games and Applications, Springer, vol. 11(3), pages 580-596, September.
    2. Stefanny Ramirez & Dario Bauso, 2023. "Dynamic Games with Strategic Complements and Large Number of Players," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 1-21, April.

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